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Properties

Label 400.3.bg.f
Level $400$
Weight $3$
Character orbit 400.bg
Analytic conductor $10.899$
Analytic rank $0$
Dimension $64$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [400,3,Mod(17,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.17"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([0, 0, 13])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.bg (of order \(20\), degree \(8\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 6 q^{5} + 4 q^{7} - 40 q^{9} - 16 q^{11} + 24 q^{13} - 82 q^{15} - 8 q^{17} + 50 q^{19} - 100 q^{21} + 48 q^{23} - 200 q^{25} - 90 q^{27} - 108 q^{31} + 260 q^{33} - 2 q^{35} - 94 q^{37} - 320 q^{39}+ \cdots - 544 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −0.811497 + 5.12359i 0 1.63338 − 4.72568i 0 4.59566 + 4.59566i 0 −17.0331 − 5.53440i 0
17.2 0 −0.583470 + 3.68389i 0 3.77915 + 3.27383i 0 −8.52562 − 8.52562i 0 −4.67107 − 1.51772i 0
17.3 0 −0.526150 + 3.32198i 0 −1.34260 + 4.81637i 0 6.50894 + 6.50894i 0 −2.19922 − 0.714570i 0
17.4 0 −0.0258720 + 0.163349i 0 −1.84061 − 4.64889i 0 −3.03617 − 3.03617i 0 8.53350 + 2.77270i 0
17.5 0 −0.00512856 + 0.0323805i 0 −4.99090 + 0.301564i 0 0.994182 + 0.994182i 0 8.55849 + 2.78082i 0
17.6 0 0.298963 − 1.88758i 0 4.96497 + 0.590862i 0 0.603173 + 0.603173i 0 5.08593 + 1.65252i 0
17.7 0 0.754209 − 4.76189i 0 −3.14938 + 3.88348i 0 −8.90117 − 8.90117i 0 −13.5472 − 4.40177i 0
17.8 0 0.898945 − 5.67572i 0 −0.177358 − 4.99685i 0 8.20346 + 8.20346i 0 −22.8462 − 7.42316i 0
33.1 0 −5.60705 − 0.888069i 0 −0.904406 − 4.91752i 0 3.60781 − 3.60781i 0 22.0908 + 7.17774i 0
33.2 0 −3.88880 − 0.615926i 0 2.05707 + 4.55724i 0 −5.11305 + 5.11305i 0 6.18393 + 2.00928i 0
33.3 0 −1.33419 − 0.211315i 0 −4.90670 + 0.961386i 0 5.78356 − 5.78356i 0 −6.82409 − 2.21728i 0
33.4 0 −0.535050 − 0.0847436i 0 2.05203 − 4.55951i 0 −4.47022 + 4.47022i 0 −8.28041 − 2.69047i 0
33.5 0 0.0475930 + 0.00753798i 0 4.31624 + 2.52389i 0 8.24576 − 8.24576i 0 −8.55730 − 2.78044i 0
33.6 0 2.61980 + 0.414935i 0 −2.28576 + 4.44694i 0 −4.29378 + 4.29378i 0 −1.86835 − 0.607064i 0
33.7 0 3.58254 + 0.567419i 0 −3.75338 − 3.30335i 0 −1.35870 + 1.35870i 0 3.95314 + 1.28445i 0
33.8 0 5.11517 + 0.810163i 0 4.93022 − 0.832424i 0 0.392224 − 0.392224i 0 16.9490 + 5.50708i 0
97.1 0 −5.60705 + 0.888069i 0 −0.904406 + 4.91752i 0 3.60781 + 3.60781i 0 22.0908 − 7.17774i 0
97.2 0 −3.88880 + 0.615926i 0 2.05707 − 4.55724i 0 −5.11305 − 5.11305i 0 6.18393 − 2.00928i 0
97.3 0 −1.33419 + 0.211315i 0 −4.90670 − 0.961386i 0 5.78356 + 5.78356i 0 −6.82409 + 2.21728i 0
97.4 0 −0.535050 + 0.0847436i 0 2.05203 + 4.55951i 0 −4.47022 − 4.47022i 0 −8.28041 + 2.69047i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.f odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.bg.f 64
4.b odd 2 1 200.3.u.b ✓ 64
25.f odd 20 1 inner 400.3.bg.f 64
100.l even 20 1 200.3.u.b ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.3.u.b ✓ 64 4.b odd 2 1
200.3.u.b ✓ 64 100.l even 20 1
400.3.bg.f 64 1.a even 1 1 trivial
400.3.bg.f 64 25.f odd 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} + 20 T_{3}^{62} + 30 T_{3}^{61} - 1175 T_{3}^{60} + 852 T_{3}^{59} - 31530 T_{3}^{58} + \cdots + 49\!\cdots\!00 \) acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display